3-D attitude representation of human joints: A standardization proposal

In view of the singularities, asymmetries and other adverse properties of existing, three-dimensional definitions for joint and segment angles, the present paper proposes a new convention for unambiguous and easily interpretable, 3-D joint angles, based on the concept of the attitude ‘vector’ as derived from Euler's theorem. The suggested standard can be easily explained to non-mathematically trained clinicians, is readily implemented in software, and can be simply related to classical Cardanic/Eulerian angles. For ‘planar’ rotations about a coordinate system's axes, the proposed convention coincides with the Cardanic convention.

The attitude vector dispenses with the ‘gimbal-lock’ and non-orthogonality disadvantages of Cardanic/Eulerian conventions; therefore, its components have better metrical properties, and they are less sensitive to measurement errors and to coordinate system uncertainties than Cardanic/Eulerian angles.

A sensitivity analysis and a physical interpretation of the proposed standard are given, and some experimental results that demonstrate its advantages.     

Can one angle be simply subtracted from another to determine range of motion in three-dimensional motion analysis?

To determine the range of motion of a joint between an initial orientation and a final orientation, it is convenient to subtract initial joint angles from final joint angles, a method referred to as the vectorial approach. However, for three-dimensional movements, the vectorial approach is not mathematically correct. To determine the joint range of motion, the rotation matrix between the two orientations should be calculated, and angles describing the range of motion should be extracted from this matrix, a method referred to as the matrical approach. As the matrical approach is less straightforward to implement, it is of interest to identify situations in which the vectorial approach leads to insubstantial errors. In this study, the vectorial approach was compared to the matrical approach, and theoretical justification was given for situations in which the vectorial approach can reasonably be used. The main findings are that the vectorial approach can be used if (1) the motion is planar (Woltring HJ. 1994. 3-D attitude representation of human joints: a standardization proposal. J Biomech 27(12): 1399–1414), (2) the angles between the final and the initial orientation are small (Woltring HJ. 1991. Representation and calculation of 3-D joint movement. Hum Mov Sci 10(5): 603–616), (3) the angles between the initial orientation of the distal segment and the proximal segment are small and finally (4) when only one large angle occurs between the initial orientation of the distal segment and the proximal segment and the angle sequence is chosen in such a way that this large angle occurs on the first axis of rotation. These findings provide specific criteria to consider when choosing the angle sequence to use for movement analysis.     

Comparison of three local frame definitions for the kinematic analysis of the fingers and the wrist

Because the hand is a complex poly-articular limb, numerous methods have been proposed to investigate its kinematics therefore complicating the comparison between studies and the methodological choices. With the objective of overcoming such issues, the present study compared the effect of three local frame definitions on local axis orientations and joint angles of the fingers and the wrist. Three local frames were implemented for each segment. The “Reference” frames were aligned with global axes during a static neutral posture. The “Landmark” frames were computed using palpated bony landmarks. The “Functional” frames included a flexion–extension axis estimated during functional movements. These definitions were compared with regard to the deviations between obtained local segment axes and the evolution of joint (Cardan) angles during two test motions. Each definition resulted in specific local frame orientations with deviations of 15° in average for a given local axis. Interestingly, these deviations produced only slight differences (below 7°) regarding flexion–extension Cardan angles indicating that there is no preferred method when only interested in finger flexion–extension movements. In this case, the Reference method was the easiest to implement, but did not provide physiological results for the thumb. Using the Functional frames reduced the kinematic cross-talk on the secondary and tertiary Cardan angles by up to 20° indicating that the Functional definition is useful when investigating complex three-dimensional movements. Globally, the Landmark definition provides valuable results and, contrary to the other definitions, is applicable for finger deformities or compromised joint rotations.     

Determination of 3D spinal kinematics without defining a local vertebral coordinate system

In this paper a method is presented to calculate Euler's angles of rotation of a body segment during locomotion without a priori defining the location of the center of rotation, and without defining a local vertebral coordinate system. The method was applied to in vivo spinal kinematics. In this method, the orientation of each segment is identified by a set of three markers. The orientation of the axes of rotation is calculated based on the average position of the markers during one stride cycle. Some restrictions and assumptions should be made. The approach is viable only when the average orientation of the anatomical axes of rotation of each spinal segment during a stride cycle coincides with the three axes of the laboratory coordinate system. Furthermore, the rotations should be symmetrical with respect to both sides of the plane of symmetry of the spinal segment, and the subject should move parallel to one axis of the laboratory coordinate system. Since in experimental conditions these assumptions will only be met approximately, errors will be introduced in the calculated angles of rotation. The magnitude of the introduced errors was investigated in a computer simulation experiment. Since the maximal errors did not exceed 0.7° in a range of misalignments up to 10° between the two coordinate systems, the approach proved to be a valid method for the estimation of spinal kinematics.     

A new non-orthogonal decomposition method to determine effective torques for three-dimensional joint rotation

This paper describes a new non-orthogonal decomposition method to determine effective torques for three-dimensional (3D) joint rotation. A rotation about a joint coordinate axis (e.g. shoulder internal/external rotation) cannot be explained only by the torque about the joint coordinate axis because the joint coordinate axes usually deviate from the principal axes of inertia of the entire kinematic chain distal to the joint. Instead of decomposing torques into three orthogonal joint coordinate axes, our new method decomposes torques into three "non-orthogonal effective axes" that are determined in such a way that a torque about each effective axis produces a joint rotation only about one of the joint coordinate axes. To demonstrate the validity of this new method, a simple internal/external rotation of the upper arm with the elbow flexed at 90 degrees was analyzed by both orthogonal and non-orthogonal decomposition methods. The results showed that only the non-orthogonal decomposition method could explain the cause-effect mechanism whereby three angular accelerations at the shoulder joint are produced by the gravity torque, resultant joint torque, and interaction torque. The proposed method would be helpful for biomechanics and motor control researchers to investigate the manner in which the central nervous system coordinates the gravity torque, resultant joint torque, and interaction torque to control 3D joint rotations.     

A joint coordinate system proposal for the study of the trapeziometacarpal joint kinematics

The International Society of Biomechanics (ISB) has recommended a standardisation for the motion reporting of almost all human joints. This study proposes an adaptation for the trapeziometacarpal joint. The definition of the segment coordinate system of both trapezium and first metacarpal is based on functional anatomy. The definition of the joint coordinate system (JCS) is guided by the two degrees of freedom of the joint, i.e. flexion-extension about a trapezium axis and abduction-adduction about a first metacarpal axis. The rotations obtained using three methods are compared on the same data: the fixed axes sequence proposed by Cooney et al., the mobile axes sequence proposed by the ISB and our alternative mobile axes sequence. The rotation amplitudes show a difference of 9 degrees in flexion-extension, 2 degrees in abduction-adduction and 13 degrees in internal-external rotation. This study emphasizes the importance of adapting the JCS to the functional anatomy of each particular joint.     

A joint coordinate system proposal for the study of the trapeziometacarpal joint kinematics

The International Society of Biomechanics (ISB) has recommended a standardisation for the motion reporting of almost all human joints. This study proposes an adaptation for the trapeziometacarpal joint.

The definition of the segment coordinate system of both trapezium and first metacarpal is based on functional anatomy. The definition of the joint coordinate system (JCS) is guided by the two degrees of freedom of the joint, i.e. flexion–extension about a trapezium axis and abduction–adduction about a first metacarpal axis. The rotations obtained using three methods are compared on the same data: the fixed axes sequence proposed by Cooney et al., the mobile axes sequence proposed by the ISB and our alternative mobile axes sequence. The rotation amplitudes show a difference of 9° in flexion–extension, 2° in abduction–adduction and 13° in internal–external rotation.

This study emphasizes the importance of adapting the JCS to the functional anatomy of each particular joint.     

Functional knee axis based on isokinetic dynamometry data: Comparison of two methods, MRI validation, and effect on knee joint kinematics

This paper compares geometry-based knee axes of rotation (transepicondylar axis and geometric center axis) and motion-based functional knee axes of rotation (fAoR). Two algorithms are evaluated to calculate fAoRs: Gamage and Lasenby's sphere fitting algorithm (GL) and Ehrig et al.'s axis transformation algorithm (SARA). Calculations are based on 3D motion data acquired during isokinetic dynamometry. AoRs are validated with the equivalent axis based on static MR-images. We quantified the difference in orientation between two knee axes of rotation as the angle between the projection of the axes in the transversal and frontal planes, and the difference in location as the distance between the intersection points of the axes with the sagittal plane. Maximum differences between fAoRs resulting from GL and SARA were 5.7° and 15.4mm, respectively. Maximum differences between fAoRs resulting from GL or SARA and the equivalent axis were 5.4°/11.5mm and 8.6°/12.8mm, respectively. Differences between geometry-based axes and EA are larger than differences between fAoR and EA both in orientation (maximum 10.6°).and location (maximum 20.8mm). Knee joint angle trajectories and the corresponding accelerations for the different knee axes of rotation were estimated using Kalman smoothing. For the joint angles, the maximum RMS difference with the MRI-based equivalent axis, which was used as a reference, was 3°. For the knee joint accelerations, the maximum RMS difference with the equivalent axis was 20°/s(2). Functional knee axes of rotation describe knee motion better than geometry-based axes. GL performs better than SARA for calculations based on experimental dynamometry.     

Use of dual Euler angles to quantify the three-dimensional joint motion and its application to the ankle joint complex

This paper presents a modified Euler angles method, dual Euler angles approach, to describe general spatial human joint motions. In dual Euler angles approach, the three-dimensional joint motion is considered as three successive screw motions with respect to the axes of the moving segment coordinate system; accordingly, the screw motion displacements are represented by dual Euler angles. The algorithm for calculating dual Euler angles from coordinates of markers on the moving segment is also provided in this study. As an example, the proposed method is applied to describe motions of ankle joint complex during dorsiflexion–plantarflexion. A Flock of Birds electromagnetic tracking device (FOB) was used to measure joint motion in vivo. Preliminary accuracy tests on a gimbal structure demonstrate that the mean errors of dual Euler angles evaluated by using source data from FOB are less than 1° for rotations and 1 mm for translations, respectively. Based on the pilot study, FOB is feasible for quantifying human joint motions using dual Euler angles approach.     

Do kinematic models reduce the effects of soft tissue artefacts in skin marker-based motion analysis? An in vivo study of knee kinematics

We investigated the effects of including kinematic constraints in the analysis of knee kinematics from skin markers and compared the result to simultaneously recorded trajectories of bone pin markers during gait of six healthy subjects. The constraint equations that were considered for the knee were spherical and revolute joints, which have been frequently used in musculoskeletal modelling. In the models, the joint centres and joint axes of rotations were optimised from the skin marker trajectories over the trial. It was found that the introduction of kinematic constraints did not reduce the error associated with soft tissue artefacts. The inclusion of a revolute joint constraint showed a statistically significant increase in the mean flexion/extension joint angle error and no statistically significant change for the two other mean joint angle errors. The inclusion of a spherical joint showed a statistically significant increase in the mean flexion/extension and abduction/adduction errors. In addition, when a spherical joint was included, a statistically significant increase in the sum of squared differences between measured marker trajectories and the trajectories of the pin markers in the models was seen. From this, it was concluded that both more advanced knee models as well as models of soft tissue artefacts should be developed before accurate knee kinematics can be calculated from skin markers.     

Measurement of the screw-home motion of the knee is sensitive to errors in axis alignment

Measurements of joint angles during motion analysis are subject to error caused by kinematic crosstalk, that is, one joint rotation (e. g., flexion) being interpreted as another (e.g., abduction). Kinematic crosstalk results from the chosen joint coordinate system being misaligned with the axes about which rotations are assumed to occur. The aim of this paper is to demonstrate that measurement of the so-called "screw-home" motion of the human knee, in which axial rotation and extension are coupled, is especially prone to errors due to crosstalk. The motions of two different two-segment mechanical linkages were examined to study the effects of crosstalk. The segments of the first linkage (NSH) were connected by a revolute joint, but the second linkage (SH) incorporated gearing that caused 15 degrees of screw-home rotation to occur with 90 degrees knee flexion. It was found that rotating the flexion axis (inducing crosstalk) could make linkage NSH appear to exhibit a screw-home motion and that a different rotation of the flexion axis could make linkage SH apparently exhibit pure flexion. These findings suggest that the measurement of screw-home rotation may be strongly influenced by errors in the location of the flexion axis. The magnitudes of these displacements of the flexion axis were consistent with the inter-observer variability seen when five experienced observers defined the flexion axis by palpating the medial and lateral femoral epicondyles. Care should be taken when interpreting small internal-external rotations and abduction-adduction angles to ensure that they are not the products of kinematic crosstalk.     

Simulation of Codman's paradox reveals a general law of motion

Codman's paradox refers to a specific pattern of motion at the shoulder joint. It asked how a mysterious axial rotation about the longitudinal axis of the arm occurred during two or three sequential arm rotations that did not involve rotation about the long-axis. The objective of this paper was to find how the mysterious axial rotation occurred in the Codman's paradox. First, Codman's paradox and Codman's rotation were defined in general situations that involved arm rotations about orthogonal axes starting from the neutral attitude as well as rotations about non-orthogonal axes and starting from non-neutral attitudes. Then a general law of motion was proposed to answer the question of the Codman's paradox, which is stated as when the long-axis of the arm performs a closed-loop motion by three sequential rotations defined as Codman's rotation, it produces an equivalent axial rotation angle about the long-axis. The equivalent axial rotation angle equals the angle of swing-the second rotation in the three sequential long-axis rotations. Validity of the proposed law of motion is demonstrated by computer simulation of various Codman's rotations. Clinical relevance of the proposed law of motion is also discussed in the paper.     

The dominant mutant Wavy auricle in blade1 disrupts patterning in a lateral domain of the maize leaf

Mature maize leaves have defined cell types along the proximal distal and medial lateral axes. The patterning events that establish these axes take place early in leaf initiation. We have identified a new dominant mutation, Wavy auricle in blade1 (Wab1), which affects patterning in both axes in a dose-dependent manner. Wab1 leaves are narrower than normal leaves and displace proximal tissues distally. We show that the proximal distal patterning defects are not due to misexpression of knox genes. Genetic analyses suggest that the action of dominant Wab1 alleles is localized to a lateral domain of the leaf, located between the midvein and the marginal domain that is determined by narrow sheath function. Thus, Wab1 defines a knox-independent pathway that affects specification of the proximal distal axis of the maize leaf. We suggest that failure to elaborate a normal lateral domain in the Wab1 leaf is responsible for disrupting patterning of the proximal distal axis.     

New mathematical definition and calculation of axial rotation of anatomical joints

In the field of joint kinematics, clinical terms such as internal-external, or medical-lateral, rotations are commonly used to express the rotation of a body segment about its own long axis. However, these terms are not defined in a strict mathematical sense. In this paper, a new mathematical definition of axial rotation is proposed and methods to calculate it from the measured Euler angles are given. The definition and methods to calculate it from the measured Euler angles are given. The definition is based on the integration of the component of the angular velocity vector projected onto the long axis of the body segment. First, the absolute axial rotation of a body segment with respect to the stationary coordinate system is defined. This definition is then generalized to give the relative axial rotation of one body segment with respect to the other body segment where the two segments are moving in the three-dimensional space. The well-known Codman's paradox is cited as an example to make clear the difference between the definition so far proposed by other researchers and the new one.     

Kinematic evaluation of the finger’s interphalangeal joints coupling mechanism—variability,flexion–extension differences,triggers, locking swanneck deformities,anthropometric correlations

The human finger contains tendon/ligament mechanisms essential for proper control. One mechanism couples the movements of the interphalangeal joints when the (unloaded) finger is flexed with active deep flexor. This study’s aim was to accurately determine in a large finger sample the kinematics and variability of the coupled interphalangeal joint motions, for potential clinical and finger model validation applications. The data could also be applied to humanoid robotic hands. Sixty-eight fingers were measured in seventeen hands in nine subjects. Fingers exhibited great joint mobility variability, with passive proximal interphalangeal hyperextension ranging from zero to almost fifty degrees. Increased measurement accuracy was obtained by using marker frames to amplify finger segment motions. Gravitational forces on the marker frames were not found to invalidate measurements. The recorded interphalangeal joint trajectories were highly consistent, demonstrating the underlying coupling mechanism. The increased accuracy and large sample size allowed for evaluation of detailed trajectory variability, systematic differences between flexion and extension trajectories, and three trigger types, distinct from flexor tendon triggers, involving initial flexion deficits in either proximal or distal interphalangeal joint. The experimental methods, data and analysis should advance insight into normal and pathological finger biomechanics (e.g., swanneck deformities), and could help improve clinical differential diagnostics of trigger finger causes. The marker frame measuring method may be useful to quantify interphalangeal joints trajectories in surgical/rehabilitative outcome studies. The data as a whole provide the most comprehensive collection of interphalangeal joint trajectories for clinical reference and model validation known to us to date.     

Expression of joint moment in the joint coordinate system

The question of using the nonorthogonal joint coordinate system (JCS) to report joint moments has risen in the literature. However, the expression of joint moments in a nonorthogonal system is still confusing. The purpose of this paper is to present a method to express any 3D vector in a nonorthogonal coordinate system. The interpretation of these expressions in the JCS is clarified and an example for the 3D joint moment vector at the shoulder and the knee is given. A nonorthogonal projection method is proposed based on the mixed product. These nonorthogonal projections represent, for a 3D joint moment vector, the net mechanical action on the JCS axes. Considering the net mechanical action on each axis seems important in order to assess joint resistance in the JCS. The orthogonal projections of the same 3D joint moment vector on the JCS axes can be characterized as "motor torque." However, this interpretation is dependent on the chosen kinematic model. The nonorthogonal and orthogonal projections of shoulder joint moment during wheelchair propulsion and knee joint moment during walking were compared using root mean squares (rmss). rmss showed differences ranging from 6 N?m to 22.3 N?m between both projections at the shoulder, while differences ranged from 0.8 N?m to 3.0 N?m at the knee. Generally, orthogonal projections were of lower amplitudes than nonorthogonal projections at both joints. The orthogonal projection on the proximal or distal coordinates systems represents the net mechanical actions on each axis, which is not the case for the orthogonal projection (i.e., motor torque) on JCS axes. In order to represent the net action at the joint in a JCS, the nonorthogonal projection should be used.     

Joint axes of rotation and body segment parameters of pig limbs

To enable a quantification of net joint moments and joint reaction forces, indicators of joint loading, this study aimed to locate the mediolateral joint axes of rotation and establish the body segment parameters of the limbs of pigs (Sus scrofa). To locate the joint axes of rotation the scapulohumeral, humeroradial, carpal complex, metacarpophalangeal, coxofemoral, femorotibial, tarsal, and metatarsophalangeal joints from 12 carcasses were studied. The joints were photographed in three positions, bisecting lines drawn at fixed landmarks with their intersection marking the joint axes of rotation. The body segment parameters, i.e. the segment mass, center of mass and moment of inertia were measured on the humerus, radius/ulna, metacarpus, forepastern, foretoe, femur, tibia, metatarsus, hindpastern, and hindtoe segments from five carcasses. The segments were weighed, and their center of mass was found by balancing them. The moments of inertia of the humerus, radius/ulna, femur and tibia were found by rotating the segments. The moments of inertia of the remaining segments were calculated. Generally, the joint axes of rotation were near the attachment site of the lateral collateral ligaments. The forelimb, with segments taken as one, was significantly lighter and shorter than the hindlimb (P < 0.001). In all segments the center of mass was located 31 to 50% distal to the proximal segment end. The segment mass decreased with distance from the trunk, as did the segment moment of inertia. The results may serve as reference on the location of the joint axes of rotation and on the body segment parameters for inverse dynamic modeling of pigs.     

Estimation of 3D rotation of femur in 2D hip radiographs

Femoral radiographs are affected by the degree of rotation of the femur with respect to the plane of projection. We aimed to determine the 3D rotation of the proximal femur in 2D radiographs. A 3D Statistical Appearance Model (SAM), which was built from CT images of cadaver proximal femurs (n=33) was randomly sampled to form a training set of 500 bones. Nineteen clinical CT images were collected for testing. All CT images were rotated to ±20° in 2° division around the shaft axis, ±10° around medial-lateral axis, and by simultaneous rotation of both axes (±16° and ±8° around shaft and medial-lateral axes). In each orientation, a 2D projection was recorded for generating a 2D SAM. The outcome parameters of the 2D SAM were used as input for a linear regression model and an artificial neural network to predict the rotation. The artificial neural network estimated the rotation more accurately than the linear regression. For artificial neural networks the mean errors were 4.0° and 2.0° around the shaft and medial-lateral axes, respectively. For an individual radiograph, the confidence interval of estimation was still relatively large. However, this method has high potential to differentiate the amount of rotations in two image sets.     

Gimbals in the insect leg

We studied the common kinematic features of the coxa and trochanter in cursorial and raptorial legs, which are the short size of the podomers, predominantly monoaxial joints, and the approximate orthogonality of adjacent joint axes. The chain coxa-trochanter with its short elements and serial orthogonality of joint axes resembles the gimbals which combine versatility and tolerance to external perturbations. The geometry of legs was studied in 23 insect species of 12 orders. Insects with monoaxial joints were selected. The joint between the trochanter and the femur (TFJ) is defined either by two vestigial condyles or by a straight anterior hinge. Direction of the joint axes in the two basal podomers was assessed by 3D measurements or by goniometry in two planes. Length of the coxa is <15% (mostly <8%) of the total length of the cursorial leg, that of the trochanter <10%. Angles between the proximal and distal joint axes in the middle coxa range from 124 to 84 degrees (mean 97+/-14 degrees ), in the trochanter (in all legs studied) from 125 to 72 degrees (mean 90+/-13 degrees ). Vectors of the distal axis in the coxa are concentrated about the normal to the plane defined by the proximal axis and the midpoint between the distal condyles. These vectors in the trochanter lie at various angles to the normal; angles are correlated with the direction of the TFJ relative to the femur. Range of reduction about the TFJ is over 60 degrees in the foreleg of Ranatra linearis, Mantispa lobata and the hind leg in Carabus coriaceus (confirming observations of previous authors), 40-60 degrees in the foreleg of Vespa crabro and in the middle one in Ammophila campestris, 10-30 degrees in other studied specimens. The special role of the trochanter in autotomy and in active propulsion in some insect groups is discussed. The majority of insects possess small trochanters and slightly movable TFJs with the joint axis laying in the femur-tibia plane. We pose the hypothesis that the TFJ damps external forces, the vectors of which lie off the femur-tibia plane, the reductor muscle acting as a spring. Thus the TFJ contributes to dynamic stability of legged locomotion.     

In vivo determination of the anatomical axes of the ankle joint complex: An optimization approach

This study investigates the feasibility of a subject-specific three-dimensional model of the ankle joint complex for kinematic and dynamic analysis of movement. The ankle joint complex was modelled as a three-segment system, connected by two ideal highe joints: the talocrural and the subtalar joint. A mathematical formulation was developed to express the three-dimensional translation and rotation between the foot and shank segments as a function of the two joint angles, and 12 model parameters describing the locations of the joint axes. An optimization method was used to fit the model parameters to three-dimensional kinematic data of foot and shank markers, obtained during test movements throughout the entire physiological range of motion of the ankle joint. The movement of the talus segment, which cannot be measured non-invasively, is not necessary for the analysis.

This optimization method was used to determine the position and orientation of the joint axes in 14 normal subjects. After optimization, the discrepancy between the best fitting model and actual marker kinematics was between 1 and 3 mm for all subjects. The predicted inclination of the subtalar joint axis from the horizontal plane was 37.4±2.7°, and the medial deviation was 18.0±16.2°. The lateral side of the talucrural axis was directed slightly posteriorly (6.8±8.1°), and inclined downward by 7.0±5.4°. These results are similar to previously reported typical results from anatomical, in vitro, studies. Reproducibility was evaluated by repeated testing of one subject, which resulted in variations of about one-fifth of the standard deviation within the group, the inclination of the subtalar joint axis was significantly correlated to the arch height and a radiographic ‘tarsal index’. It is concluded that this optimization method provides the opportunity to incorporate inter-individual anatomical differences into kinematic and dynamic analysis of the ankle joint complex. This allows a more functional interpretation of kinematic data, and more realistic estimates of internal forces.